[Paper Review] Bayesian inference for general Gaussian graphical models with application to multivariate lattice data

Adrian Dobra, Alex Lenkoski+1 more|arXiv (Cornell University)|May 21, 2010

Bayesian Methods and Mixture Models26 references4 citations

TL;DR

This paper introduces a novel reversible jump MCMC sampler for Bayesian inference in multivariate and matrix-variate Gaussian graphical models (GGMs) with G-Wishart priors, enabling efficient estimation and model determination on both decomposable and non-decomposable graphs. The method avoids computationally intensive marginal likelihood calculations and enables flexible, sparse conditionally autoregressive (CAR) models for multivariate lattice data, with applications to spatial data and cancer mortality mapping.

ABSTRACT

We introduce efficient Markov chain Monte Carlo methods for inference and model determination in multivariate and matrix-variate Gaussian graphical models. Our framework is based on the G-Wishart prior for the precision matrix associated with graphs that can be decomposable or non-decomposable. We extend our sampling algorithms to a novel class of conditionally autoregressive models for sparse estimation in multivariate lattice data, with a special emphasis on the analysis of spatial data. These models embed a great deal of flexibility in estimating both the correlation structure across outcomes and the spatial correlation structure, thereby allowing for adaptive smoothing and spatial autocorrelation parameters. Our methods are illustrated using simulated and real-world examples, including an application to cancer mortality surveillance.

Motivation & Objective

• To develop a scalable Bayesian inference framework for general Gaussian graphical models (GGMs) that supports both decomposable and non-decomposable graphs.
• To eliminate the need for computationally expensive marginal likelihood calculations in model determination by using a novel reversible jump MCMC algorithm.
• To extend the connection between conditionally autoregressive (CAR) models and GGMs for multivariate lattice data, enabling adaptive smoothing and spatial correlation estimation.
• To provide a unified, efficient sampling strategy for matrix-variate GGMs that jointly models row and column conditional independence structures.
• To demonstrate the method’s utility through applications in spatial statistics, including state-level SAT scores and cancer mortality surveillance.

Proposed method

• Proposes a new Metropolis-Hastings algorithm for sampling from the G-Wishart distribution on arbitrary graphs using Cholesky decomposition, avoiding clique enumeration.
• Develops a reversible jump MCMC sampler that jointly updates the row and column precision matrices, conditional independence graphs, and auxiliary variables to resolve non-identifiability in matrix-variate normal models.
• Employs a block Gibbs sampling scheme that sequentially updates the mean vector μ, centered random effects X̃, row precision matrix KR, column graph GC, column precision matrix KC, and auxiliary variable z.
• Uses conditional priors based on matrix-variate normal distributions and G-Wishart constraints to encode conditional independence in both row and column structures.
• Applies a Metropolis-Hastings step with a normal proposal to update individual elements of the centered random effects matrix X̃ when the full conditional is non-standard.
• Integrates the G-Wishart prior into a hierarchical model framework that supports both continuous and discrete multivariate lattice data via generalized linear models.

Experimental results

Research questions

• RQ1 How can Bayesian model determination in general GGMs be performed efficiently without computing marginal likelihoods?
• RQ2 Can a single MCMC algorithm be designed to handle both decomposable and non-decomposable graphs in GGMs using the G-Wishart prior?
• RQ3 How can the connection between CAR models and GGMs be fully exploited in multivariate lattice data with non-decomposable neighborhood structures?
• RQ4 What is the performance of the proposed sampler in estimating spatial and cross-outcome correlation structures in real-world multivariate spatial data?
• RQ5 Can the method provide adaptive smoothing and flexible spatial correlation estimation in applications like cancer mortality mapping?

Key findings

• The proposed MCMC sampler avoids the need to compute normalizing constants of the G-Wishart posterior, significantly improving computational efficiency over existing methods.
• The method achieves accurate inference on non-decomposable graphs by leveraging Cholesky-based sampling, circumventing the need for clique enumeration.
• The sampler enables the construction of sparse, multivariate CAR models that jointly estimate spatial correlation and cross-variable dependence structures.
• In the cancer mortality application, the model successfully identified meaningful spatial patterns and conditional independence structures across states.
• The simulation study confirms the method’s robustness in recovering true graph structures and precision matrices under various sparsity and correlation regimes.
• The algorithm demonstrates strong mixing and convergence properties in high-dimensional settings, outperforming prior approaches in computational speed and scalability.